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Properties

Label 1764.4.t.a
Level $1764$
Weight $4$
Character orbit 1764.t
Analytic conductor $104.079$
Analytic rank $0$
Dimension $16$
CM no
Inner twists $8$

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Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [1764,4,Mod(521,1764)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1764, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([0, 3, 1]))
 
N = Newforms(chi, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1764.521");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1764 = 2^{2} \cdot 3^{2} \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 1764.t (of order \(6\), degree \(2\), not minimal)

Newform invariants

comment: select newform
 
sage: f = N[0] # Warning: the index may be different
 
gp: f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(104.079369250\)
Analytic rank: \(0\)
Dimension: \(16\)
Relative dimension: \(8\) over \(\Q(\zeta_{6})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{16} + \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{16} + 42x^{12} + 1139x^{8} + 26250x^{4} + 390625 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{29}]\)
Coefficient ring index: \( 2^{24}\cdot 3^{16} \)
Twist minimal: no (minimal twist has level 252)
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{15}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_{11} q^{5}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_{11} q^{5} + ( - \beta_{4} - \beta_{2}) q^{11} + ( - \beta_{14} + \beta_{12}) q^{13} + ( - \beta_{15} + \beta_{13} + \beta_{7}) q^{17} + ( - 2 \beta_{14} + 2 \beta_{10} - 3 \beta_{9}) q^{19} + ( - 2 \beta_{6} - 2 \beta_{3} + 5 \beta_{2}) q^{23} + ( - 4 \beta_{8} - 5 \beta_1) q^{25} + (\beta_{6} + 10 \beta_{4}) q^{29} + ( - 6 \beta_{12} + 6 \beta_{9}) q^{31} + (6 \beta_{5} - 64 \beta_1 - 64) q^{37} + ( - 3 \beta_{13} - 29 \beta_{11} + 29 \beta_{7}) q^{41} + (5 \beta_{8} - 5 \beta_{5} + 4) q^{43} + (3 \beta_{15} - 32 \beta_{11}) q^{47} + ( - 18 \beta_{4} - 13 \beta_{3} - 18 \beta_{2}) q^{53} + ( - 4 \beta_{14} - 7 \beta_{12}) q^{55} + (2 \beta_{15} - 2 \beta_{13} - 12 \beta_{7}) q^{59} + ( - 10 \beta_{14} + \cdots - 20 \beta_{9}) q^{61}+ \cdots + ( - 15 \beta_{14} - \beta_{12}) q^{97}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q+O(q^{10}) \) Copy content Toggle raw display \( 16 q + 40 q^{25} - 512 q^{37} + 64 q^{43} + 2336 q^{67} - 1792 q^{79} - 2688 q^{85}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{16} + 42x^{12} + 1139x^{8} + 26250x^{4} + 390625 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( 42\nu^{12} + 1139\nu^{8} + 47838\nu^{4} + 390625 ) / 711875 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( -771\nu^{12} - 71757\nu^{8} - 878169\nu^{4} - 20238750 ) / 1423750 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 801\nu^{14} + 174267\nu^{10} + 912339\nu^{6} + 21026250\nu^{2} ) / 35593750 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( -3\nu^{12} - 6993 ) / 2278 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 3\nu^{14} - 150189\nu^{2} ) / 28475 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( -378\nu^{14} - 10251\nu^{10} - 53667\nu^{6} - 3515625\nu^{2} ) / 2093750 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 1028 \nu^{15} - 21000 \nu^{13} - 95676 \nu^{11} - 569500 \nu^{9} - 2594642 \nu^{7} + \cdots - 373281250 \nu ) / 88984375 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( -6567\nu^{14} - 228939\nu^{10} - 7479813\nu^{6} - 172383750\nu^{2} ) / 17796875 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 2861 \nu^{15} - 7600 \nu^{13} - 94537 \nu^{11} - 1053575 \nu^{9} - 4682429 \nu^{7} + \cdots - 857984375 \nu ) / 88984375 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( - 3304 \nu^{15} - 1850 \nu^{13} + 384982 \nu^{11} + 797300 \nu^{9} + 6202994 \nu^{7} + \cdots - 297718750 \nu ) / 88984375 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 4778 \nu^{15} - 10500 \nu^{13} + 95676 \nu^{11} - 284750 \nu^{9} + 2594642 \nu^{7} + \cdots - 453593750 \nu ) / 88984375 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( - 7597 \nu^{15} + 54475 \nu^{13} - 189074 \nu^{11} + 1053575 \nu^{9} - 9364858 \nu^{7} + \cdots + 113000000 \nu ) / 88984375 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( - 734 \nu^{15} + 13050 \nu^{13} - 45828 \nu^{11} + 204350 \nu^{9} + 252724 \nu^{7} + \cdots + 87125000 \nu ) / 5234375 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( 19642 \nu^{15} + 45950 \nu^{13} + 769964 \nu^{11} + 398650 \nu^{9} + 12405988 \nu^{7} + \cdots + 707875000 \nu ) / 88984375 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( - 30614 \nu^{15} + 59400 \nu^{13} - 389538 \nu^{11} - 3473950 \nu^{9} + 2148154 \nu^{7} + \cdots - 611968750 \nu ) / 88984375 \) Copy content Toggle raw display

Display \(\nu^j\) in terms of \(\beta_i\)

\(\nu\)\(=\) \( ( 2\beta_{14} - 2\beta_{12} - 12\beta_{11} - \beta_{10} - 2\beta_{9} + 6\beta_{7} ) / 72 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( -2\beta_{6} - 3\beta_{5} - 2\beta_{3} ) / 18 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( -4\beta_{15} - \beta_{14} + 2\beta_{13} - 14\beta_{12} - 26\beta_{11} + 2\beta_{10} + 28\beta_{9} - 26\beta_{7} ) / 72 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( 2\beta_{4} + 2\beta_{2} + 63\beta_1 ) / 3 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( 10 \beta_{15} - 37 \beta_{14} - 20 \beta_{13} + 28 \beta_{12} + 122 \beta_{11} - 37 \beta_{10} + \cdots - 244 \beta_{7} ) / 72 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( -51\beta_{8} + 134\beta_{6} + 51\beta_{5} ) / 18 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( 84 \beta_{15} - 166 \beta_{14} + 84 \beta_{13} - 338 \beta_{12} + 684 \beta_{11} + 83 \beta_{10} + \cdots - 342 \beta_{7} ) / 72 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( -28\beta_{2} - 257\beta _1 - 257 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( - 840 \beta_{15} - 929 \beta_{14} + 420 \beta_{13} + 662 \beta_{12} + 1374 \beta_{11} + \cdots + 1374 \beta_{7} ) / 72 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( ( 267\beta_{8} + 4378\beta_{3} ) / 18 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( ( 2278 \beta_{15} + 4111 \beta_{14} - 4556 \beta_{13} + 10892 \beta_{12} + 1886 \beta_{11} + \cdots - 3772 \beta_{7} ) / 72 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( ( -2278\beta_{4} - 6993 ) / 3 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( ( 11390 \beta_{15} + 31786 \beta_{14} + 11390 \beta_{13} + 36554 \beta_{12} + 37084 \beta_{11} + \cdots - 18542 \beta_{7} ) / 72 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( ( -100126\beta_{6} + 20661\beta_{5} - 100126\beta_{3} ) / 18 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( ( - 86352 \beta_{15} + 120787 \beta_{14} + 43176 \beta_{13} - 17482 \beta_{12} + 292962 \beta_{11} + \cdots + 292962 \beta_{7} ) / 72 \) Copy content Toggle raw display

Display \(\beta_i\) in terms of \(\nu^j\)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1764\mathbb{Z}\right)^\times\).

\(n\) \(785\) \(883\) \(1081\)
\(\chi(n)\) \(-1\) \(1\) \(1 + \beta_{1}\)

Embeddings

For each embedding \(\iota_m\) of the coefficient field, the values \(\iota_m(a_n)\) are shown below.

For more information on an embedded modular form you can click on its label.

comment: embeddings in the coefficient field
 
gp: mfembed(f)
 
Label   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
521.1
2.07637 + 0.829861i
1.75687 + 1.38326i
−0.829861 + 2.07637i
1.38326 − 1.75687i
0.829861 − 2.07637i
−1.38326 + 1.75687i
−2.07637 − 0.829861i
−1.75687 − 1.38326i
2.07637 − 0.829861i
1.75687 − 1.38326i
−0.829861 − 2.07637i
1.38326 + 1.75687i
0.829861 + 2.07637i
−1.38326 − 1.75687i
−2.07637 + 0.829861i
−1.75687 + 1.38326i
0 0 0 −7.66648 − 13.2787i 0 0 0 0 0
521.2 0 0 0 −7.66648 − 13.2787i 0 0 0 0 0
521.3 0 0 0 −1.10680 − 1.91704i 0 0 0 0 0
521.4 0 0 0 −1.10680 − 1.91704i 0 0 0 0 0
521.5 0 0 0 1.10680 + 1.91704i 0 0 0 0 0
521.6 0 0 0 1.10680 + 1.91704i 0 0 0 0 0
521.7 0 0 0 7.66648 + 13.2787i 0 0 0 0 0
521.8 0 0 0 7.66648 + 13.2787i 0 0 0 0 0
1097.1 0 0 0 −7.66648 + 13.2787i 0 0 0 0 0
1097.2 0 0 0 −7.66648 + 13.2787i 0 0 0 0 0
1097.3 0 0 0 −1.10680 + 1.91704i 0 0 0 0 0
1097.4 0 0 0 −1.10680 + 1.91704i 0 0 0 0 0
1097.5 0 0 0 1.10680 − 1.91704i 0 0 0 0 0
1097.6 0 0 0 1.10680 − 1.91704i 0 0 0 0 0
1097.7 0 0 0 7.66648 − 13.2787i 0 0 0 0 0
1097.8 0 0 0 7.66648 − 13.2787i 0 0 0 0 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 521.8
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
7.b odd 2 1 inner
7.c even 3 1 inner
7.d odd 6 1 inner
21.c even 2 1 inner
21.g even 6 1 inner
21.h odd 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1764.4.t.a 16
3.b odd 2 1 inner 1764.4.t.a 16
7.b odd 2 1 inner 1764.4.t.a 16
7.c even 3 1 252.4.f.a ✓ 8
7.c even 3 1 inner 1764.4.t.a 16
7.d odd 6 1 252.4.f.a ✓ 8
7.d odd 6 1 inner 1764.4.t.a 16
21.c even 2 1 inner 1764.4.t.a 16
21.g even 6 1 252.4.f.a ✓ 8
21.g even 6 1 inner 1764.4.t.a 16
21.h odd 6 1 252.4.f.a ✓ 8
21.h odd 6 1 inner 1764.4.t.a 16
28.f even 6 1 1008.4.k.d 8
28.g odd 6 1 1008.4.k.d 8
84.j odd 6 1 1008.4.k.d 8
84.n even 6 1 1008.4.k.d 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
252.4.f.a ✓ 8 7.c even 3 1
252.4.f.a ✓ 8 7.d odd 6 1
252.4.f.a ✓ 8 21.g even 6 1
252.4.f.a ✓ 8 21.h odd 6 1
1008.4.k.d 8 28.f even 6 1
1008.4.k.d 8 28.g odd 6 1
1008.4.k.d 8 84.j odd 6 1
1008.4.k.d 8 84.n even 6 1
1764.4.t.a 16 1.a even 1 1 trivial
1764.4.t.a 16 3.b odd 2 1 inner
1764.4.t.a 16 7.b odd 2 1 inner
1764.4.t.a 16 7.c even 3 1 inner
1764.4.t.a 16 7.d odd 6 1 inner
1764.4.t.a 16 21.c even 2 1 inner
1764.4.t.a 16 21.g even 6 1 inner
1764.4.t.a 16 21.h odd 6 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{8} + 240T_{5}^{6} + 56448T_{5}^{4} + 276480T_{5}^{2} + 1327104 \) acting on \(S_{4}^{\mathrm{new}}(1764, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{16} \) Copy content Toggle raw display
$3$ \( T^{16} \) Copy content Toggle raw display
$5$ \( (T^{8} + 240 T^{6} + \cdots + 1327104)^{2} \) Copy content Toggle raw display
$7$ \( T^{16} \) Copy content Toggle raw display
$11$ \( (T^{4} - 414 T^{2} + 171396)^{4} \) Copy content Toggle raw display
$13$ \( (T^{4} + 5088 T^{2} + 3875328)^{4} \) Copy content Toggle raw display
$17$ \( (T^{8} + 16368 T^{6} + \cdots + 37903417344)^{2} \) Copy content Toggle raw display
$19$ \( (T^{8} + \cdots + 2767192326144)^{2} \) Copy content Toggle raw display
$23$ \( (T^{8} + \cdots + 88\!\cdots\!16)^{2} \) Copy content Toggle raw display
$29$ \( (T^{4} + 83124 T^{2} + 1700572644)^{4} \) Copy content Toggle raw display
$31$ \( (T^{8} + \cdots + 85\!\cdots\!24)^{2} \) Copy content Toggle raw display
$37$ \( (T^{4} + 128 T^{3} + \cdots + 661106944)^{4} \) Copy content Toggle raw display
$41$ \( (T^{4} - 328560 T^{2} + 22577275008)^{4} \) Copy content Toggle raw display
$43$ \( (T^{2} - 8 T - 20684)^{8} \) Copy content Toggle raw display
$47$ \( (T^{8} + \cdots + 17\!\cdots\!24)^{2} \) Copy content Toggle raw display
$53$ \( (T^{8} + \cdots + 12\!\cdots\!96)^{2} \) Copy content Toggle raw display
$59$ \( (T^{8} + \cdots + 39\!\cdots\!44)^{2} \) Copy content Toggle raw display
$61$ \( (T^{8} + \cdots + 49\!\cdots\!00)^{2} \) Copy content Toggle raw display
$67$ \( (T^{4} - 584 T^{3} + \cdots + 5186304256)^{4} \) Copy content Toggle raw display
$71$ \( (T^{4} + 331308 T^{2} + 17756628516)^{4} \) Copy content Toggle raw display
$73$ \( (T^{8} + \cdots + 28\!\cdots\!04)^{2} \) Copy content Toggle raw display
$79$ \( (T^{4} + 448 T^{3} + \cdots + 4768731136)^{4} \) Copy content Toggle raw display
$83$ \( (T^{4} - 1430592 T^{2} + 328711292928)^{4} \) Copy content Toggle raw display
$89$ \( (T^{8} + \cdots + 58\!\cdots\!84)^{2} \) Copy content Toggle raw display
$97$ \( (T^{4} + 816096 T^{2} + 149346427392)^{4} \) Copy content Toggle raw display
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